# §25.4 Reflection Formulas

For $s\neq 0,1$,

 25.4.1 $\zeta\left(1-s\right)=2(2\pi)^{-s}\cos\left(\tfrac{1}{2}\pi s\right)\Gamma% \left(s\right)\zeta\left(s\right),$ ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$: gamma function, $\zeta\left(\NVar{s}\right)$: Riemann zeta function, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$: cosine function and $s$: complex variable Keywords: reflection formula Source: Apostol (1976, (12), p. 259) Referenced by: §25.10(i), §5.17 Permalink: http://dlmf.nist.gov/25.4.E1 Encodings: TeX, pMML, png See also: Annotations for §25.4 and Ch.25
 25.4.2 $\zeta\left(s\right)=2(2\pi)^{s-1}\sin\left(\tfrac{1}{2}\pi s\right)\Gamma\left% (1-s\right)\zeta\left(1-s\right).$ ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$: gamma function, $\zeta\left(\NVar{s}\right)$: Riemann zeta function, $\pi$: the ratio of the circumference of a circle to its diameter, $\sin\NVar{z}$: sine function and $s$: complex variable Keywords: reflection formula Source: Apostol (1976, (13), p. 259) A&S Ref: 23.2.6 Permalink: http://dlmf.nist.gov/25.4.E2 Encodings: TeX, pMML, png See also: Annotations for §25.4 and Ch.25

Equivalently,

 25.4.3 $\xi\left(s\right)=\xi\left(1-s\right),$ ⓘ Symbols: $\xi\left(\NVar{s}\right)$: Riemann’s $\xi$-function and $s$: complex variable Keywords: reflection formula Source: Apostol (1976, p. 260) Permalink: http://dlmf.nist.gov/25.4.E3 Encodings: TeX, pMML, png See also: Annotations for §25.4 and Ch.25

where $\xi\left(s\right)$ is Riemann’s $\xi$-function, defined by:

 25.4.4 $\xi\left(s\right)=\tfrac{1}{2}s(s-1)\Gamma\left(\tfrac{1}{2}s\right)\pi^{-s/2}% \zeta\left(s\right).$ ⓘ Defines: $\xi\left(\NVar{s}\right)$: Riemann’s $\xi$-function Symbols: $\Gamma\left(\NVar{z}\right)$: gamma function, $\zeta\left(\NVar{s}\right)$: Riemann zeta function, $\pi$: the ratio of the circumference of a circle to its diameter and $s$: complex variable Keywords: definition Source: Apostol (1976, p. 260) Permalink: http://dlmf.nist.gov/25.4.E4 Encodings: TeX, pMML, png See also: Annotations for §25.4 and Ch.25

For $s\neq 0,1$ and $k=1,2,3,\dots$,

 25.4.5 $(-1)^{k}{\zeta}^{(k)}\left(1-s\right)=\frac{2}{(2\pi)^{s}}\sum_{m=0}^{k}\sum_{% r=0}^{m}\genfrac{(}{)}{0.0pt}{}{k}{m}\genfrac{(}{)}{0.0pt}{}{m}{r}\left(\Re% \left(c^{k-m}\right)\cos\left(\tfrac{1}{2}\pi s\right)+\Im\left(c^{k-m}\right)% \sin\left(\tfrac{1}{2}\pi s\right)\right){\Gamma}^{(r)}\left(s\right){\zeta}^{% (m-r)}\left(s\right),$

where

 25.4.6 $c\equiv-\ln\left(2\pi\right)-\tfrac{1}{2}\pi\mathrm{i}.$ ⓘ Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $\equiv$: equals by definition, $\mathrm{i}$: imaginary unit, $\ln\NVar{z}$: principal branch of logarithm function and $c$ Keywords: definition Source: Apostol (1985a, p. 223) Referenced by: §25.6(ii) Permalink: http://dlmf.nist.gov/25.4.E6 Encodings: TeX, pMML, png See also: Annotations for §25.4 and Ch.25